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The standard form of a quadratic equation is given by the equation ax 2 + bx + c = 0, where a ≠ 0. It means the quadratic equation has a variable raised to 2 as the greatest power term. The word "quadratic" is originated from the word "quad" and its meaning is "square". If it’s not correct, go back and see if you can find where you went wrong and work through the steps again.Before going to learn about solving quadratic equations, let us recall a few facts about quadratic equations. In order for the quadratic equation to equal zero at least one of the factors must be zero, so you can solve it by setting each factor equal to zero and solving - like you would for a linear equation.Ĭheck that the answer(s) you have obtained is/are actually correct, by substituting the value back into the original equation and evaluating it. You can solve like this as follows:įactorise the quadratic equation, as detailed in the Factorising section of this module. This isn’t always easy or even possible, but if you notice that the equation can be relatively easily factorised, this can be a quick method of solving it.
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For example, \(\sqrt &Īnother way you can solve a quadratic equation is by factorising the equation using two sets of brackets.
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Rearrange the equation so that \(ax^2\) is by itself on the left hand side of the equation, by adding or subtracting the constant \(c\) from both sides, as applicable like you would do for a linear equation.ĭivide through both sides of the equation by \(a\) (assuming \(a \neq 1\)) like you would do for a linear equation. You can solve equations like this as follows: Probably the easiest type of quadratic equation to solve is one where \(b = 0\) i.e. This page outlines three different techniques. There are various ways of solving quadratic equations, depending on the nature of the equation. The second image is an example of where the parabola intersects the \(x\) axis twice, and hence there are two solutions, and finally the third image is an example of where the parabola does not intersect the \(x\) axis at all - and hence there are no solutions. The first image below is an example of where the parabola only intersects the \(x\) axis once, and hence there is only one solution (this is actually a plot of \(y = x^2\), although no scales are included on the axes). As seen below these take the form of parabolas, and the places at which these parabolas intersect the \(x\) (horizontal) axis are the solutions to the equation (as this is when the equation is equal to \(0\)). To understand why this is the case, it can be helpful to look at the graphs of some quadratic equations. While this quadratic equation has only one solution, note that often there will be two solutions, and sometimes there will be no (real) solutions at all. So the solution to this quadratic equation is \(0\). In this case you should be able to see that the value of the variable \(x\) must be \(0\), as only \(0^2 = 0\). The most basic quadratic equation occurs when \(a = 1\), \(b = 0\) and \(c = 0\), in which case we have: These are referred to as coefficients of the equation. Where \(x\) is a variable and \(a\), \(b\) and \(c\) represent known numbers such that \(a \neq 0\) (if \(a = 0\) then the equation is linear). In particular, the page covers the following (use the drop-down menu above to jump to a different section as required):Ī quadratic equation, or second degree equation, is an algebraic equation of the form: This page explains some techniques for solving quadratic equations.